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A Formal Solution for Wave Propagation in Rectangular Waveguide with an Inserted Nonlinear Dielectric Slab
Salvatore Caorsi, Andrea Massa, Matteo Pastorino
To cite this version:
Salvatore Caorsi, Andrea Massa, Matteo Pastorino. A Formal Solution for Wave Propagation in Rectangular Waveguide with an Inserted Nonlinear Dielectric Slab. Journal de Physique III, EDP Sciences, 1995, 5 (1), pp.43-57. �10.1051/jp3:1995109�. �jpa-00249294�
Classificatidn
Ph_vsics Abstracts
41,10H 03.40K 42.20G 42.658
A Formal Solution for Wave Propagation in Rectangular
Waveguide with an Inserted Nonlinear Dielectric Slab
Salvatore Caorsi, Andrea Massa and Matteo Pastorino
Department of Biophysical and Electronic Engineering, University of Genoa, Via all'opera Pia, I IA, 1-16145 Genova, Italy
(Received 24 March 1994, re»ised 25 July J994, accepted 20 October 1994)
Rdsum4,- Dans cet article nous prdsentons une solution formelle en sdrie pour ddterminer la distribution du champ dlectromagndtique dans
un guide d'onde rectangulaire dans Iequel une
plaque didlectrique non Iindaire a dtd insdrde. La solution est ddveloppde en termes de la fonction de Green pour Ies guides d'ondes rectangulaires dont Ies parois sont parfaitement conductrices.
Nous supposons que Ie guide d'onde est excitd
en mode TEjo h la frdquence fo. La mdthode tient compte de la gdndration de modes supdrieurs dvoluant h la frdquence fn et h des frdquences
diffdrentes. Nous supposons aussi que la plaque non Iindaire est isotropique et non homogdne (mdme sans champ appliqud). L'artide montre comment la solution du probldme (c'est-h-dire Ie calcul des coefficients de la sdrie) peut dtre rdduite h la solution d'un systbme d'dquations intdgrales coupldes. La solution formelle est ddrivde pour un guide d'onde infini et un guide d'onde
court-circuitd.
Abstract. A formal series solution is presented for the electromagnetic field distribution inside a
rectangular waveguide in which a nonlinear dielectric slab has been inserted. The solution is
developed in terms of the Green function for rectangular waveguides with perfectly conductive walls. We assume the waveguide to be excited in the TEjo mode at a frequency fo. The approach takes into account the higher-order mode generation at fo and at different frequencies. We also
assume the nonlinear slab to be isotropic and inhomogeneous (even without any impressed field).
The paper show~ how the problem solution (I.e., the computation of the series coefficients) can be reduced to the solution of a system of coupled integral equations. The iormal solution is derived for
two cases : an infinite waveguide and a short-circuited waveguide.
1. Introduction
This paper deals with nonlinear electromagnetic wave propagation in a guiding structure. In
recent years, wave propagation in nonlinear media has been extensively investigated. There
exists a vast literature on this topic ; we refer the reader, for example, to some works [1-4] and to the references cited therein. Considerable efforts have been devoted to studying partial
differential equations for nonlinear propagation, to defining interesting phenomena, like
soliton formation and decay [5], and shock waves [6], and to describing a large number of
applications for which it is impossible to provide a complete list of reference~. Analytical,
@Les Editions de Physique 1995
44 JOURNAL DE PHYSIQUE III N°
variational and approximate methods have been devised and numerical techniques are
becoming increasingly important. In the past, some numerical results in the field of nonlinear
wave propagation have constituted a strong stimulus for further theoretical studies of very great interest.
However, nonlinear wave propagation is in general considered for propagation media of infinite extent, both in free space and inside guiding structures (conductive and dielectric waveguides, planar structures, etc.). Very few works have addressed the interaction of
electromagnetic waves with nonlinear bodies of limited dimensions, the scattering from which had to be evaluated. In this paper, we discuss the case of a bounded nonlinear slab inserted in a
rectangular waveguide. We assume the waveguide to be excited in its fundamental mode, the
TEjo mode, at a frequency fo. The slab interfaces with air through cylindrical surfaces whose
axes are parallel to the E-field polarization. The dielectric permittivity of the slab is assumed to be dependent on the internal electric field. In addition, we assume that the operator that links the nonlinear dielectric permittivity to the electric field vector is such as not to modify the scalar nature of the permittivity (consequently, it does not produce depolarization in the wave field). Moreover, the slab is inhomogeneous not only due to its nonlinearity, but even when no
e-m- field is applied (I.e., the linear part of the relative dielectric permittivity itself is inhomogeneous). The remaining part of the waveguide is empty (or filled with a linear homogeneous dielectric). Nonmagnetic materials are assumed for all media and the conductive
walls of the waveguide are assumed to be perfect conductors. In the following section, we
develop a formal series solution for the electromagnetic waves inside the waveguide in terms
of the Green function for rectangular waveguides. This solution takes into account the
generation of direct and reflected higher-order modes at frequency fo and at harmonic
frequencies. Then, we describe how the use of a specified nonlinear operator results in a
numerical problem solution in which all the coefficients of the series expansion for the
electromagnetic fields in the various waveguide regions are given as the solution of a system of nonlinear coupled integral equations. The case of a particular nonlinearity (whose highest
order is proportional to the power) is detailed. Finally, the possibility of solving numerically
the resulting system of integral equations is discussed. In the paper, the mathematical formulation of the approach is presented for the cases of an infinite waveguide and of a semi- infinite waveguide closed by a short circuit and loaded with the described nonlinear slab.
2. The Electromagnetic Field Problem
Let us consider Figure I, in which three regions are shown. For these regions, the following mathematical relations for the electromagnetic quantities hold.
2,I. REGIONS I AND III. Under the assumptions made in the Introduction, in region1
(= < =~ (,<)), the total time-dependent electric field vector e~~~(r, t) given by the sum of the incident and the reflected fields) is polarized along the y axis and is independent of the
v coordinate e~~~(r, ii
= e(~~(.<, =, t) y.
As is well-known, e(,~ Jo, z, t) can be expressed as a Fourier series with a fundamental
+m
pulsation wo = 2 wfo e(,'~(>~, =, t
= ~j E(~'(,x, z e~"~'~' where E('~(>., =), which satisfies
,,=-~
" "
V)E)~'j.t., z) + wj
~z~ ej E)'~(.t., z)
=
0 (w~~ = nw~l (1)
iS given by
El,,( (-t, =
=
I Siu ")
.tlhjj,j, e '~'~'~~°
+ A,,,,, e~'~'~'~"° (2)
j,>
z
x
b
e
o
III
~~~'~~'"ZU- ~i(~)
d3~r,t),~,n., d~(r,t)r.fl.
~2~~~
Fig. I. -Rectangular waveguide and nonlinear dielectric slab.
where h~~
=
0 form # I and n # I, h~~ = I form
=
I and n
=
I (we assume a unit amplitude
for the fundamental incident mode), and
~g~ 2
Y~,~ " ~°~ /~0 ~i ~~~
a
The corresponding vector component of the magnetic field is related to the electric field by the
Maxwell equation and is given by
~ll)~~ ~) l ~ j~(1)(~ ~)~_ ~ £(11(~ ~)~) (~)
~ ~
jW~/L~ 3z ~~ ~ 3~t '~ ~
and its x component can be expressed as
H)~l(,;, z) z (~,~~ sin "~"
x[h~~~ e~~?~"~~ A~,~ e~~~~~~] (5)
" ~
46 JOURNAL DE PHYSIQUE III N°
where Yj~' is the wave admittance equal to
~'~"
In region III (z m z2(x)), the total electric w~
~o
field vector e~~~(r, t) (given by the transmitted field) is still polarized in the y direction : e~~~(r, t) = e)~~(x, z, t) y. By analogy to the field in region I, we can expand e)~~(.<, z, t) in
Fourier series whose n-th term satisfies
V)E)~l(x, z) + wj ~zo e~ E(~~(x, z, t )
=
0 (6)
and is given by :
+ ~ j~)
E)~l(x, = = z D~~ sin '~ " x e~~?~" ~ (7)
m
~
where
~ ~ 2
yjfl (~~
w j lLo E~
a
The magnetic field vector is given by a relation corresponding to (4) (valid for region Ii, and its
~ component can be expressed as
H)~~(x, z = (
Y$$~D~~ sin '~" x e~~?~'~~~ (9)
~
m =1
~
where
y13)
~
~~~'~
mn
~°>1 /L0
2.2. REGION II. Under the hypothesis that the total electric field, e~~J(r, t), is still y-
polarized (e~~~(r, t)
= e(,~J(>, =, t y), in region II (zj(,i < = < z~ (x)) (I.e., the nonlinear slab), the following homogeneous wave equation holds
V)e(,~~(x, z, t) ~zo ~~ E~(>., z, t)e(,~~(.<, z, t
=
0 (10)
~
where E~ix, z, t) is given by :
E~(.i~, =, t eo[e~j (>, z) + e~~ o (e~~~(x, z, t )) (I I)
where e~j(x, =) is the linear part of the relative dielectric permittivity, which can be
inhomogeneous itself (when no field is applied) o (e~~~(x, z, t)) is a nonlinear operator and
e~~ is a constant parameter. The nonlinear operator is assumed to fulfil the constraint of not
modifying the scalar nature of the dielectric permittivity (isotropic medium). We also assume o (e'~~(x,z, t)) to be a time-periodic function. Then, we can write: e(~~(.<,z,t)=
+~ +~
jj Ef~(x, = e~~"°' and o (e~~~(.<, =, t))
= z o,j(.i, z) e~""° The product of these quan-
,< ~
"
,j=-m
tities can also be written in Fourier series as
+~ +~ +~
z E)?'(x, z) e~"""~ z o~~(x, z) e~'~"~'~ z V'j~(i, z) e~'"~~ (12)
n
i,=-m n,=-~ h=-~
where:
~'~~" °~ i~
~[~~(-;, z) o~(,,., z) 6h
~~~" ~'"-~
mn
(13)
where 6(,,
= 1, if m + n
=
h, and 6(,~
=
0, otherwise. Now, for each frequency w,,
= NW o,
relation (10) can be rewritten as :
il~l~))~(.t, Z) + W~ /L0 ~0 ~21("' ~~~~j~~~' ~~ ~ ~°~ ~° ~° ~~~ ~'~~~' ~~ ~ ~~~~
and the magnetic field vector becomes
~l~~l~' ~)
#
l 3
~~~~
~~~ ~~ ~~ ~~ ~~~
~~~ X ~~)~"'~) Z) ~~~~
,,
The boundary conditions for region II are
nxel~~(x,z,t)(~~ ~,j=nxe~~~(x,z,t)(~~~~,~ (16)
n x e~~~(.x, =, t)(-~~,j,j
= n x e~~~(,<, z, t)(-~
~,~ (17)
n x h~~~(.<, =, t)(z=z~(,j ~ n X h~~~(~, ~, ~~(z=z l'J ~~~~
n x h~~J(.<, z, t (, ~,j = n x h~~J(.<, z, t ~
-~j,~. (19)
For each n, the formal solution of equation (14) can be obtained by considering the term
WI ~zo eo e~~ P~(x, =) as an equivalent source term, according to the equivalence principle
for electromagnetic fields [7] :
vjJ~jji(.;, zi + w,i ~n ~o ~~j (>., =i El[~(,i, z) = W,I~£o Eo e22 *n(.<, z) ~~°~
It should be noted that the terms V',~(.<, =) generally depend on the mixing of all the field components, E(,(~(,<, =), for any m. Under the above conditions, we express the field solution
as E()~ = E))~~°~+E()~~P~ where E()~~°J denotes the solution of the homogeneous wave
equation (corresponding to (20)) in region II (with E(.i, z)
= eo E~jO, z)) and E(~~lPl is a
particular solution (dependent on P~(.<, =)) that can be expressed in terms of tit Green
function for rectangular waveguides
a =~(,
E(~ ~ w j
~z~ F~ e~~ V',~(~i', =' G,,(.r, z/~<', z' da.' d=' (2
"
0 ~j,1
where G,,(,i, z/x'. =' ) is given by [81
G,,(x, z/x', z' ) = (
~
sin '~ " x sin '~ " x' e~~?~~ ° (22)
n,
Y$~ ~ ~
where yj$~ w,)~z~ E~ E~j 16~" It should be stressed that, unlike in the linear
a
~
case, this is just a formal solution, as in relation (21), for any n, E(,~~~~ depends (through
48 JOURNAL DE PHYSIQUE III N°
V'~(x, z)) on all the field components, E))~ (total field), for any j, including j
= n. Now, since the solution for E)~~1°1 is given by
+~ ~g~ <21~
~ >2)~
E)~l~°~(x, z
=
z sin
.<[Bf)e~~ ?"'~ + C$~~
e ~?~"' (23)
~
n,
~
the electric field in region II can be forrnally expressed as :
El')~(~' ~ ~
i ~i~ j .~ lB(I + Bmn (~)l ~~~~~~~ + ICC) + Cmn(Z)1 ~~~~~"~ (24)
~~l ~
where, of course, B~~(z) and C~,~(z) are unknown coefficients still dependent on E(~l(x, ?) and, in general, on all the other field components. Such coefficients can be defined
as
~ a = ~~ ,~,
B,nn(z) = £°
o lLo E~(yf~~)~ E22 o V',>(.~', z') sin -.<' e~~?~" ° da' dz' (25)
=~(,)
~
~~~~° ~~ ~° ~°~Y~#)
Ez~ j~ ~~~'l ~
° z
~~~
'
~~~ ~~~ ~ x' e~'?I,~z' ~_~, ~~, Moreover, the x components of the magnetic field are given by
H(~~(,<, z =
(
Yj)~~ sin'~"
x
[B$)
+ B~,,~(z)] e~~~~"~ + j ~ B~,~(z)
~
m =1
~ Ymn ~~
ict~ + Cm, (z)i e~'~~~/ J ( I Cmiiz
)1<?7J
where
Y~~~ "
/~) (28)
,j o
From relation (24), one can deduce again that the harmonic components depend on the
others i,ia the nonlinear coefficients V'~(x, z). These coefficients are responsible for the mixing
of modes (at fo and at dil~ferent frequencies), which, in a linear medium, propagate without interactions (if somewhere generated). They can be derived directly from relation (13), once the nonlinear operator o (e~~l(.x, z, t )) in it has been defined.
3. Parallelepipedal Dielectric Slab
In order to show how to handle equations (24-28), we consider the simple case in which the inserted nonlinear slab is a parallelepiped. We set zj(x) = 0 and z~(.<) = zo (Fig. 2a). The
coupling coefficients, B~,~(z) and C~~(z) of the nonlinear slab become
B~~~(z
= w ~zo E~ yj(~l)~ ' E~~
~
V'~(x', z') sin '~" x' e~~~~~~ dJ.' dz' (29)
~ 0
~
'~~
a co
~q~ ~ <2,~,
C~~(z)
= WI~zo E~ y((1)~ e~~ 0 z V',,(x', =' ) sin x' e ~?~'~ dx' dz' (30)
a
a
~ )[[))(~
E~~
j~
~~~ ~~~~'~~~
, ten.
ii
~'
i~
a)
1 di(r,t)me.
/-
e~(r,t)
re,i.
~d~'
e~
r~
bl
~~g. 2. ~a) Infinite Waveguide containing a nonlinear slab parallepipedal in shape. b) Waveguide
terrrinated by an ideal short circuit.
Two cases are described. In the first case, the waveguide is assumed to be infinite (it is of
course equivalent to a waveguide terrninated by a matched load (at all frequencies) in the
second case, we assume the waveguide to be short-circuited (I.e., terminated by a perfectly
conductive wall).
3.I. INFINITE WAVEGUIDE. By imposing, at the interfaces zj(x) 0 and z~(x) = zo, the
boundary conditions, we obtain the explicit dependence of each linear coefficient A~~, Bj$), C($~, and D~~~ (relations (2), (7), and (23)) on the nonlinear coefficients B~~(z) and C~~,~(z) (all still unknown), calculated for zj(x) 0 and z~(x) zo. In particular, for each
n and m, a system of equations is obtained by applying the boundary conditions. Considering
the linear coefficients of this system of equations as unknowns and the nonlinear coefficients as
unknown terms, for each n and m, the system turns out to be made up of four equations and
four unknowns. The explicit solution of these systems (e.g., by the Cramer method) gives the following relations (the related trivial steps are omitted)
~
~
~
~~ ~~j~~,y)(~zo ~ B~ (z~) (Y)(~- l'~~~~ ~ ~~~~~ ~
~~ ~
j '~)~ y)(~[Cji(0) (l"1 ~ ~~~
j~,~ a,_
(Y))~ Y))~) (e~~~" ~° e'~" '°) (31)
2 Y)(~(Y))~ + Y))~)~?~'~~°) (32)
50 JOURNAL DE PHYSIQUE Ill N°
~(0)_ ~-jy)(lzo(~ ~Q)j~~(1))2 ~~(2))2j~-JY~l'zo ~~ (~)( (2) (1))2~-JY((~zo ~
~~ det[Sjjl ~~ ~~ ~~ ~~ ~ ~'~~ ~'~~
+ 2 Y)(~(Y1)~ Y))~) e~~?'~
~") (33) Djj
=
~ y)(~[Bjj(0)(y((~+ y))~) + Cjj(z~) (y))~- y)~~) + 2 y)~~l (34) det [S~ii
An>n " ~j )/~ e~~~~~~~° jc
n,,j (o ( y]/n~ + y))/) / ~fl~° + Bnjn(20 ) yj$~ y$(/) e~ ~~/
~° (35)
~~
CCl
~ ~~~
~
~
~~~"~~
(Bmn(0)1(yfi)~ (y)i)/)~l /~~~~
+ cmn(20) (yil y$I)~ e~~~il~°)
n,~
(37)
Dmn ~
~
~ y$>~lB>n»(0) (y$[~ + Y]$~) + Cn>n(20) (y$~ y$)i~Jl ~~~~
e njn
where det [Sjii and det [S~~] are given by
dot lsiil
= e~~?~"~~l(Y)(~ + Y)i~)~e~?'I'°° (y)(~- Y))~)~ e~~?'~~°°j (39) d~~ lsmn1
"
~~~~~~~~ l(y$I + yil)~ ~~~~"~° (y)/n~ y)#)~ ~ ~~~~~~~~l (40)
3.2. SHORT-CIRCUITED WAVEGUIDE. In the case where the waveguide is terminated by a
short circuit, we assume z w zo, with the additional boundary conditions (Fig. 2b) nxe'~l(x,=,t)=0 for
z=zo. (41)
As in the previous case, the application of the boundary conditions gives the relations
between the linear and nonlinear coefficients
~~ '~'°
j~~ jyll'zu_
B~~(zo) Y(~~e ~~~~
2 Y)i
iiijc~~(0) Yii
~~~=_ e
~ ~~jYil
121~ >2)_
<Y)(~- Y))~)e'?~~ ~° (Y))~+ Yl(~)e~?~~ -°j (42)
~10) ~ ~~~ (2i (1))~'Y)~~zu ~ ~ ~~ ~~121_ iii)~-JY()~°u ~
~~ det[Sjj] ~~ ~'~~ ~'~~ ~~ ° ~~ ~'~~
~~12,-
+ 2 y()~e ~~ ~°] (43)
~~ j
~ _,~121~
~
C)1 12~_
~ lBj~(=o) (y)i~ + y)j~) e ~~ ~~ + C11 (0) (y))~- y)]I)- e~~?" ~° + det [Sjjj
~2~~
+ 2 y)(~e~~?'~ ~"] (44)
~~~ ~~
~-jYmn ij ~~~~
~~
j ~))
jcm,,(0) Yii~e~~~"'° ~~"~~°
B©I
"
~~ ~
lC
mn
(0 (yl$~ y$(/) / ~i ~° + Bmn (20 yIn~
yn[>~) e ~
~~~~°
(46)
n,~
CInS~ -
~~~
s~~j iBmn(zo) ~Yii~ + Yi?)e~?~/° Cmn~°)
(YIn~n~ YIn?)e~?~~°i (47)
where
dot js~~j
=
(y))~+ y()~)/?'~~~°
+ (Y))~- Y))~)
e
~~~~~~~ ~~~~
d~~ lsmnl
~ (y$)>~+ y$~)/~~~~~
+ (y))/~ y]>[~)~~'~~~~~~~ (49)
4. Formal Series Solutions
The formal series solutions can now be completed in both cases. For the infinite waveguide,
relations (31-38) express the linear coefficients A~~~, B)$), C)$~~, D~,,~ as functions of the unknown nonlinear coefficients B~~ (z) and C
~~
(z ), calculated for zj(,<)
=
0 and z~(x) = z~. By substituting relations (31-38) into (24), we can express E(,)~(x, z) only as a function of
B~~(01, B~,~(?o), C~~(0), and C~,,~(zo). If the nonlinear operator o (e~~~(r, t)) is specified,
from relation (13) we derive the term P,,(x, z) as a function of B~~(z) and C~,,(z). Finally,
substitution into (25) and (26) reduces the problem to the solution of the following system of nonlinear integral equations
a z
~m>i(Z) ~ WI /~0 ~0(Yj>~~ ~22 0 0 ~'fi(~m»(Z~), ~m>i(Z~), A', Z~) X
~q ~ >2~~.
x sin x' e~~?~"'~ dJ' dz' (50)
a
a =~
~mn(Z)
~ WI /L0 ~0(Y~~/~ ~22 ~'>i(~m>i(Z'), ~nm(Z'), X', Z') X
0 =
x sin '~ " ~r' e~'?~~'~ da.' dz' (51)
a
~~~
where the dependence of $r,,(.i, =) on B~~~(=) and C~,,~(z) is indicated.
For the short-circuited waveguide, the same results can be obtained by considering
relations (42-47), by substituting into (24), and by using (25) and (26). Relations formally equal to (50) and (51) can then be derived.
5. Choice of the Nonlinearity
In this section, under the assumptions made in Section I, we consider a particular choice for the nonlinear operator (Eq. (I I)), which is assumed to be given by :
o (e~~'(~, z, t))
= o~(e)~~(,x, =, t1)~ + oj e(~~(x, z, t (52)
where Hi and o~ are known constants. Relation (52) corresponds to a nonlinearity for the
relative dielectric permittivity truncated at the second-order term (dependence on the field
